Time-Frequency Characteristic Analysis of Time Series: Fourier Transform and Inverse Transform

In time series analysis, we can investigate the time-frequency characteristics of sequential data. This includes studying Fourier transforms and their inverse operations, along with implementations of fast Mellin transforms with inverse procedures, and short-time discrete Fourier transforms (STFT) - techniques commonly implemented using functions like fft(), ifft(), and spectrogram() in signal processing libraries. These methods enable extraction of instantaneous frequency information through spectral analysis algorithms. Furthermore, we can analyze various time-frequency distribution visualizations including Born-Jordan distributions computed using Cohen's class distribution formulas, Butterworth-filtered time-frequency representations with adjustable cutoff parameters, and Choi-Williams distributions that employ exponential kernels to reduce cross-term interference. These graphical representations provide intuitive observation of instantaneous frequency variations through matrix-based plotting techniques and contour mapping functions.